+ def one_basis(self):
+ """
+ Return the basis-element-index of this algebra's unit element.
+ """
+ return 0
+
+
+ def one(self):
+ """
+ Return the multiplicative identity element of this algebra.
+
+ The superclass method computes the identity element, which is
+ beyond overkill in this case: the algebra identity should be our
+ first basis element. We implement this via :meth:`one_basis`
+ because that method can optionally be used by other parts of the
+ category framework.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
+ ....: random_eja)
+
+ EXAMPLES::
+
+ sage: J = RealCartesianProductEJA(5)
+ sage: J.one()
+ e0 + e1 + e2 + e3 + e4
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: A.one()
+ f0
+ sage: A.one().superalgebra_element()
+ e0 + e1 + e2 + e3 + e4
+
+ TESTS:
+
+ The identity element acts like the identity::
+
+ sage: set_random_seed()
+ sage: J = random_eja().random_element().subalgebra_generated_by()
+ sage: x = J.random_element()
+ sage: J.one()*x == x and x*J.one() == x
+ True
+
+ The matrix of the unit element's operator is the identity::
+
+ sage: set_random_seed()
+ sage: J = random_eja().random_element().subalgebra_generated_by()
+ sage: actual = J.one().operator().matrix()
+ sage: expected = matrix.identity(J.base_ring(), J.dimension())
+ sage: actual == expected
+ True
+ """
+ if self.dimension() == 0:
+ return self.zero()
+ else:
+ return self.monomial(self.one_basis())
+
+
+ def natural_basis_space(self):
+ """
+ Return the natural basis space of this algebra, which is identical
+ to that of its superalgebra.
+
+ This is correct "by definition," and avoids a mismatch when the
+ subalgebra is trivial (with no natural basis to infer anything
+ from) and the parent is not.
+ """
+ return self.superalgebra().natural_basis_space()
+
+